Use impleit differentiation to determine \( \frac{d y}{d x} \) ror the equation \( x^{4} \sqrt{y}-9 x^{6}-6-c^{4} \)

Given the equation \[ x^4\sqrt{y} - 9x^6 - 6 - c^4 = 0, \] we want to use implicit differentiation to find \(\frac{dy}{dx}\). **Step 1. Differentiate both sides with respect to \(x\):** Differentiate term by term. 1. For the first term, \(x^4\sqrt{y}\), use the product rule: \[ \frac{d}{dx}\left(x^4\sqrt{y}\right) = \frac{d}{dx}\left(x^4\right)\sqrt{y} + x^4\frac{d}{dx}\left(\sqrt{y}\right). \] We have \[ \frac{d}{dx}\left(x^4\right)=4x^3. \] To differentiate \(\sqrt{y} = y^{1/2}\), apply the chain rule: \[ \frac{d}{dx}\left(y^{1/2}\right) = \frac{1}{2}y^{-1/2}\frac{dy}{dx} = \frac{1}{2\sqrt{y}}\frac{dy}{dx}. \] Therefore, \[ \frac{d}{dx}\left(x^4\sqrt{y}\right) = 4x^3\sqrt{y} + \frac{x^4}{2\sqrt{y}}\frac{dy}{dx}. \] 2. The derivative of \(-9x^6\) is \[ -9 \cdot 6x^5 = -54x^5. \] 3. Both \(-6\) and \(-c^4\) are constants, so their derivatives are 0. Thus, differentiating the entire equation with respect to \(x\) gives: \[ 4x^3\sqrt{y} + \frac{x^4}{2\sqrt{y}}\frac{dy}{dx} - 54x^5 = 0. \] **Step 2. Solve for \(\frac{dy}{dx}\):** Isolate the term containing \(\frac{dy}{dx}\): \[ \frac{x^4}{2\sqrt{y}}\frac{dy}{dx} = 54x^5 - 4x^3\sqrt{y}. \] Multiply both sides by \(\frac{2\sqrt{y}}{x^4}\) to solve for \(\frac{dy}{dx}\): \[ \frac{dy}{dx} = \frac{2\sqrt{y}}{x^4}\left(54x^5 - 4x^3\sqrt{y}\right). \] Simplify by distributing: \[ \frac{dy}{dx} = \frac{108x^5\sqrt{y} - 8x^3y}{x^4}. \] Divide each term by \(x^4\): \[ \frac{dy}{dx} = \frac{108x^5\sqrt{y}}{x^4} - \frac{8x^3y}{x^4} = 108x\sqrt{y} - \frac{8y}{x}. \] Thus, the derivative is \[ \frac{dy}{dx} = 108x\sqrt{y} - \frac{8y}{x}. \]